Teichmuller Theory and Handle Addition for Minimal Surfaces

In this work is presented a family that generalizes Bour’s, Enneper’s, Richmond’s and Henneberg’s minimal surfaces. First of all is introduced a generalization for the well-known Bour’s family $$B_m$$ by including a new index r, determining a new bigger family $$B_{m r}$$ , which includes Bour’s, Enneper’s and Richmond’s families of minimal surfaces. After that, realizing some similarities between the Enneper–Weierstrass data of the Henneberg’s minimal surface and the $$B_{m r}$$ family, it is established a new family of minimal surfaces $$H_{m r}$$ that includes the classical Henneberg minimal surface. Finally, it is presented a bigger family $$C_ {m r}^{\kappa h}, \ m,r \in {\mathbb {R}}, \ \kappa \in {\mathbb {C}}$$ which includes as particular cases $$B_{m r}$$ and $$H_{m r}$$ families. Consequently, catenoid, helicoid, Enneper’s minimal surfaces, Bour’s minimal surfaces, Richmond’s minimal surfaces and Henneberg’s minimal surfaces are just some elements of the set of $$C_ {m r}^{\kappa h}$$ surfaces.

In this paper we have proved several approximation theorems for the family of minimal surfaces in \({\mathbb{R}}^{3}\) that imply, among other things, that complete minimal surfaces are dense in the space of all minimal surfaces endowed with the topology of Ck convergence on compact sets, for any \(k \in {\mathbb{N}}\) .

Second order contact of minimal surfaces

: A minimal surface is the surface of least area bounded by a given closed curve. In three dimensional space these surface are realized, for reasonably simple curves, by soap films spanning wire loops. This paper obtains results related to the theorem that the normals to a complete minimal surface, not a plane, are everywhere dense. Examples: l. The same theorem holds for two-dimensional surfaces in Euclidean spaces of dimension three or more. 2. Theorems connecting total curvature, Euler characteristic, and number of boundary components. 3. Theorems about the normals and tangent planes to minimal surfaces, including the capacity of the set of directions omitted. (Author)

In this paper we find approximate solutions of certain Riemann-Hilbert boundary value problems for minimal surfaces in $\mathbb{R}^n$ and null holomorphic curves in $\mathbb{C}^n$ for any $n\ge 3$. With this tool in hand we construct complete conformally immersed minimal surfaces in $\mathbb{R}^n$ which are normalized by any given bordered Riemann surface and have Jordan boundaries. We also furnish complete conformal proper minimal immersions from any given bordered Riemann surface to any smoothly bounded, strictly convex domain of $\mathbb{R}^n$ which extend continuously up to the boundary; for $n\ge 5$ we find embeddings with these properties.

In this chapter we continue and substantially expand the study of the Gauss map of minimal surfaces. The main new result of this chapter is that every natural candidate is the Gauss map of a conformal minimal surface in Rn. We also discuss the value distribution theory of the Gauss map of complete minimal surfaces of finite total curvature.

Thanks to the works of Callahan, Costa, Hoffman, Karcher, Meeks, Rosenberg, Wei, etc.—see for example [1],[2],[3],[5],[8],[11],[12],[16],[20]—, we dispose now of a large number of properly embedded minimal surfaces in the euclidean space IR other than the classical examples. All those surfaces can be viewed as minimal surfaces with finite total curvature properly embedded in complete flat three manifolds. The most basic invariants associated to a surface of this type are its topology and its periodicity. It is an interesting problem to decide if the simplest examples—like the catenoid, the helicoid, the Scherk’s surfaces or the Riemann example—can be characterized in terms of some of these invariants. In the non-periodic case, there are two important uniqueness theorems in this direction: the first one, obtained by Schoen [17], characterizes the catenoid as the only complete minimal surface embedded

We study complete minimal surfaces in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document} with finite total curvature and embedded planar ends. After conformal compactification via inversion, these yield examples of surfaces stationary for the Willmore bending energy W:=14∫|H→|2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {W}: =\\frac{1}{4} \\int |\\vec H|^2$$\\end{document}. In codimension one, we prove that the W\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {W}$$\\end{document}-Morse index for any inverted minimal sphere or real projective plane with m such ends is exactly m-3=W4π-3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m-3=\\frac{\\mathcal {W}}{4\\pi }-3$$\\end{document}. We also consider several geometric properties—for example, the property that all m asymptotic planes meet at a single point—of these minimal surfaces and explore their relation to the W\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {W}$$\\end{document}-Morse index of their inverted surfaces.

We consider unbranched Willmore surfaces in Euclidean space that arise as inverted complete minimal surfaces with embedded planar ends. Several statements are proven about upper and lower bounds on the Morse Index—the number of linearly independent variational directions that locally decrease the Willmore energy. We in particular compute the Index of a Willmore sphere in the three‑space. This Index is $m-d$, where $m$ is the number of ends of the corresponding complete minimal surface and $d$ is the dimension of the span of the normals at the $m$-fold point. The dimension $d$ is either two or three. For $m=4$, we prove that $d=3$. In general, we show that there is a strong connection of the Morse Index to the number of logarithmically growing Jacobi fields on the corresponding minimal surface.

We show that under the hypothesis of bounded Gaussian curvature, certain minimal surfaces are in fact of finite total curvature. We can then answer the following version of a conjecture of Nitsche (J. Math. Mech. 11 (1962), 295) under the hypothesis of bounded Gaussian curvature: Conjecture. Let M 2 ⊂ R 3 {M^2} \subset {{\mathbf {R}}^3} be a complete minimal surface such that for some height function H H , the level sets are (compact) Jordan curves. Then M M is a catenoid.

The Weierstrass data are derived—from geometric assumptions—for a family of screw-motion-invariant minimal surfaces asymptotic to the helicoid. The period problem for these data is solved numerically and the the surfaces are approximated using adaptive mesh methods. These simulations give strong evidence that the family exists, is continuous, consists of embedded surfaces, and limits to the genus-one helicoid.

Let Ω be a bounded C2 domain in ℝn and ϕ ∂Ω → ℝm be a continuous map. The Dirichlet problem for the minimal surface system asks whether there exists a Lipschitz map f : Ω → ℝm with f|∂Ω = ϕ and with the graph of f a minimal submanifold in ℝn+m. For m = 1, the Dirichlet problem was solved more than 30 years ago by Jenkins and Serrin [12] for any mean convex domains and the solutions are all smooth. This paper considers the Dirichlet problem for convex domains in arbitrary codimension m. We prove that if ψ : ¯Ω → ℝm satisfies 8nδ supΩ |D2ψ| + √2 sup∂Ω |Dψ| < 1, then the Dirichlet problem for ψ|∂Ω is solvable in smooth maps. Here δ is the diameter of Ω. Such a condition is necessary in view of an example of Lawson and Osserman [13]. In order to prove this result, we study the associated parabolic system and solve the Cauchy‐Dirichlet problem with ψ as initial data. © 2003 Wiley Periodicals, Inc.

A Schoen theorem for minimal surfaces in [formula omitted

For each $k\geq 3$ , we construct a $1$ -parameter family of complete properly Alexandrov-embedded minimal surfaces in the Riemannian product space $\mathbb {H}^2\times \mathbb {R}$ with genus $1$ and k embedded ends asymptotic to vertical planes. We also obtain complete minimal surfaces with genus $1$ and $2k$ ends in the quotient of $\mathbb {H}^2\times \mathbb {R}$ by an arbitrary vertical translation. They all have dihedral symmetry with respect to k vertical planes, as well as finite total curvature $-4k\pi $ . Finally, we provide examples of complete properly Alexandrov-embedded minimal surfaces with finite total curvature with genus $1$ in quotients of $\mathbb {H}^2\times \mathbb {R}$ by the action of a hyperbolic or parabolic translation.

In [19], A. King states the following conjecture: Any smooth complete toric variety has a tilting bundle whose summands are line bundles. The goal of this paper is to prove King’s conjecture for the following types of smooth complete toric varieties: (i) Any d-dimensional smooth complete toric variety with splitting fan. (ii) Any d-dimensional smooth complete toric variety with Picard number ≤2. (iii) The blow up of any smooth complete minimal toric surface at T-invariants points.